The profile drag coefficient is constant and the lift coefficient is dependant upon the angle of attack of the aircraft. Therefore as the angle of attack changes, so does the lift coefficient and also the drag coefficient.

Differentiating [tex]C_D[/tex] with respect to [tex]C_L[/tex] does find the minimum drag, however this happens when [tex]C_L = 0[/tex]. Since the question states that the aircraft is in straight and level flight, it is obvious that [tex]C_L \neq 0[/tex]. The minimum drag during flight would be when the performance of the aircraft (i.e. the lift-to-drag ratio) is at its maximum, i.e:

[tex]\frac{d^2C_L}{dC_D^2} < 0[/tex]

Drag is defined as:

[tex]C_D=C_{D_0}+kC_L^2[/tex]

Dividing through by [tex]C_L[/tex] gives:

[tex]\frac{C_D}{C_L}=\frac{C_{D_0}}{C_L}+kC_L[/tex]

Since this is the reciprocal of the performance (lift-to-drag ratio), finding the minimum of this function will be the same as finding the maximum of the performance, which will give the minimum drag of the aircraft.